Question: The values of $a$, $b$, $c$ and $d$ are 1, 2, 3 and 4, but not necessarily in that order. What is the largest possible value of the sum of the four products $ab$, $bc$, $cd$ and $da$?
Solution: Seeing pairwise products, we consider  \[
(a+b+c+d)^2=a^2+b^2+c^2+d^2+2(ab+ac+ad+bc+bd+cd),
\]so \[
ab+bc+cd+da=\frac{(a+b+c+d)^2-a^2-b^2-c^2-d^2}{2}-(ac+bd).
\]Since the fraction on the right-hand side does not depend on how the values of $a$, $b$, $c$, and $d$ are assigned, we maximize $ab+bc+cd+da$ by minimizing $ac+bd$.  Checking the three distinct values for $ac+bd$, we find that $1\cdot4+2\cdot3=10$ is its minimum value.  Therefore, the largest possible value of $ab+bc+cd+da$ is $$\frac{(1+2+3+4)^2-1^2-2^2-3^2-4^2}{2}-10=\boxed{25}.$$